Let $a,b,c$ be non-negative real numbers such that $a+b\leq c+1, b+c\leq a+1$ and $c+a\leq b+1.$ Show that \[a^2+b^2+c^2\leq 2abc+1.\]

Find all positive integers $n$ such that $\phi(n)$ is the fourth power of some prime.

A function $f : I \to \mathbb R$, defined on an interval $I$, is called concave if $f(\theta x + (1 - \theta)y) \geq \theta f(x) + (1 - \theta)f(y)$ for all $x, y \in I$ and $0 \leq \theta \leq 1$. Assume that the functions $f_1, \ldots , f_n$, having all nonnegative values, are concave. Prove that the function $(f_1f_2 \cdots f_n)^{1/n}$ is concave.

For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions: $(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$. $(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$. Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.

Let $P$ be a regular $2n$-sided polygon. A [b]rhombus-ulation[/b] of $P$ is dividing $P$ into rhombuses such that no two intersect and no vertex of any rhombus is on the edge of other rhombuses or $P$. (a) Prove that number of rhombuses is a function of $n$. Find the value of this function. Also find the number of vertices and edges of the rhombuses as a function of $n$. (b) Prove or disprove that there always exists an edge $e$ of $P$ such that by erasing all the segments parallel to $e$ the remaining rhombuses are connected. (c) Is it true that each two rhombus-ulations can turn into each other using the following algorithm multiple times? Algorithm: Take a hexagon -not necessarily regular- consisting of 3 rhombuses and re-rhombus-ulate the hexagon. (d) Let $f(n)$ be the number of ways to rhombus-ulate $P$. Prove that:\[\Pi_{k=1}^{n-1} ( \binom{k}{2} +1) \leq f(n) \leq \Pi_{k=1}^{n-1} k^{n-k} \]

Let be a function $ \xi:\mathbb{R}\to\mathbb{R} $ of class $ C^{\infty } $ such that $ \left| \frac{d^n\xi }{dx^n} \left( x_0 \right) \right|\le 1=\frac{d\xi}{dx}(0) , $ for any real numbers $ x_0, $ and all natural numbers $ n, $ and let be the function $ h:\mathbb{C}\longrightarrow\mathbb{C} , h(z)=1+\sum_{n\in\mathbb{N}} \left(\frac{z^n}{n!}\cdot\frac{d^n\xi }{dx^n} \left( 0 \right)\right) . $ [b]a)[/b] Show that $ h $ is well-defined and analytic. [b]b)[/b] Prove that $ h\bigg|_{\mathbb{R}} =\xi\bigg|_{\mathbb{R}} . $ [b]c)[/b] Demonstrate that $$ \frac{d}{dt}\left( \frac{\xi }{\cos} \right)\left( t_0 \right) =4\sum_{p\in\mathbb{Z}}\frac{(-1)^p\xi\left( \frac{(1+2p)\pi}{2} \right)}{\left( (1+2p)\pi -2t_0\right)^2} , $$ for any $ t_0\in\left( -\frac{\pi }{2} ,\frac{\pi }{2} \right) $ and that $$ \sum_{p\in\mathbb{Z}} \frac{(-1)^p\left(\xi\left( \frac{(1+2p)\pi}{2} \right)\right)^2}{1+2p} =\frac{\pi }{2} . $$ [b]d)[/b] Deduce that $ \xi\left( \frac{(1+2p)\pi}{2} \right)=(-1)^p, $ for any integer $ p, $ and that $$ \frac{d}{dt}\left( \frac{\xi }{\cos} \right)\left( t_0 \right) =\frac{d}{dt}\left( \frac{\sin }{\cos} \right)\left( t_0 \right) , $$ for any $ t_0\in\left( -\frac{\pi }{2} ,\frac{\pi }{2} \right) . $ [b]e)[/b] Conclude that $ \xi\bigg|_\mathbb{R} =\sin\bigg|_\mathbb{R} . $

Let $a$ be a real number. Suppose the function $f(x) = \frac{a}{x-1} + \frac{1}{x-2} + \frac{1}{x-6}$ defined in the interval $3 < x < 5$ attains its maximum at $x=4$. Find the value of $a.$

Let $ n$ be a positive integer. Given an integer coefficient polynomial $ f(x)$, define its [i]signature modulo $ n$[/i] to be the (ordered) sequence $ f(1), \ldots , f(n)$ modulo $ n$. Of the $ n^n$ such $ n$-term sequences of integers modulo $ n$, how many are the signature of some polynomial $ f(x)$ if a) $ n$ is a positive integer not divisible by the square of a prime. b) $ n$ is a positive integer not divisible by the cube of a prime.

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ that satisfy the equation: $ f\left(\left(f\left(x\right)\right)^2 \plus{} f\left(y\right)\right) \equal{} xf\left(x\right) \plus{} y$ for any two real numbers $ x$ and $ y$.

Let $a$ and $b$ be positive integers, $b<a$, such that $a^3+b^3+ab$ is divisible by $ab(a-b)$. Prove that $ab$ is a perfect cube.

Let $ f:[1,\infty )\longrightarrow (0,\infty ) $ be a continuous function satisfying the following properties: $ \text{(i)}\exists\lim_{x\to\infty } \frac{f(x)}{x}\in\overline{\mathbb{R}} $ $ \text{(ii)}\exists\lim_{x\to\infty } \frac{1}{x}\int_1^x f(t)dt\in\mathbb{R}. $ [b]a)[/b] Show that $ \lim_{x\to\infty } \frac{f(x)}{x}=0. $ [b]b)[/b] Prove that $ \lim_{x\to\infty } \frac{1}{x^2}\int_1^x f^2(t)dt=0. $

For a permutation $\pi$ of $\{1,2,3,\ldots,n\}$, let $\text{Inv}(\pi)$ be the number of pairs $(i,j)$ with $1 \leq i < j \leq n$ and $\pi(i) > \pi(j)$. [list=1] [*] Given $n$, what is $\sum \text{Inv}(\pi)$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$? [*] Given $n$, what is $\sum \left(\text{Inv}(\pi)\right)^2$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?[/list] [i]Brian Hamrick.[/i]

Let $ f$ be a function with the following properties: (i) $f(1) \equal{} 1$, and (ii) $ f(2n) \equal{} n\times f(n)$, for any positive integer $ n$. What is the value of $ f(2^{100})$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2^{99} \qquad \textbf{(C)}\ 2^{100} \qquad \textbf{(D)}\ 2^{4950} \qquad \textbf{(E)}\ 2^{9999}$

A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$. For how many $x$ we might have $P(x)=x$?